Hi there.
I found the information in a book called 'An Alphabetical Dictionary of Coats of Arms forming an extensive Ordinary of British Armorials ...' by J. W. Papworth (1874) that is in my hearldic library.
To get more information I think that you would have to spend some $s £s or €s by making official enquiries at the offices of the British Heraldic Authorities (England/Wales in London, Ireland in Dublin and Edinburgh in Scotland) who have far more extensive records, many of which have not yet been released to the public.
Do John Walkers in Dublin still exist? If so, you might try and contact them and see if anything exists in their records. The surname is more common in Ireland and Scotland that it is in England.
Regards,
GJKS